// Numbas version: exam_results_page_options
{"name": "Change of variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Evaluate the integral below:</p>\n<p style=\"padding-left: 30px;\">\\(\\int\\int_R\\var{k2}(x^2+y^2)dxdy\\)</p>\n<p style=\"padding-left: 30px;\">where \\(R\\) is the region of the plane enclosed by&nbsp;\\(\\var{a1}\\le \\var{k}xy\\le \\var{b1}\\)&nbsp; &nbsp;and&nbsp; &nbsp;\\(\\var{a2}\\le x^2-y^2\\le \\var{b2}\\).</p>\n<p style=\"padding-left: 90px;\"></p>", "advice": "<p>\\(\\int\\int_R\\var{k2}(x^2+y^2)dxdy\\)</p>\n<p>\\(R\\) is the region of the plane enclosed by&nbsp;\\(\\var{a1}\\le \\var{k}xy\\le \\var{b1}\\)&nbsp; &nbsp;and&nbsp; &nbsp;\\(\\var{a2}\\le x^2-y^2\\le \\var{b2}\\)</p>\n<p>Let \\(u=\\var{k}xy\\)&nbsp; and let&nbsp; \\(v=x^2-y^2\\)</p>\n<p>Limits:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}\\le u\\le \\var{b1}\\)&nbsp; and&nbsp; \\(\\var{a2}\\le v\\le \\var{b2}\\)</p>\n<p>Jacobian:</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{vmatrix} \\frac{du}{dx}&amp;\\frac{du}{dy}\\\\ \\frac{dv}{dx}&amp;\\frac{dv}{dy}\\\\ \\end{vmatrix}=\\begin{vmatrix} \\var{k}y&amp;\\var{k}x\\\\2x&amp;-2y\\\\\\end{vmatrix}=\\simplify{-2*{k}}y^2-\\simplify{2*{k}}x^2=-\\simplify{2*{k}}(x^2+y^2)\\)</p>\n<p style=\"padding-left: 30px;\">\\(The\\,Jacobian\\,is\\,the\\,absolute\\,value\\,of\\,the\\,determinant:J=\\simplify{2*{k}}(x^2+y^2)\\)</p>\n<p style=\"padding-left: 30px;\">\\(dxdy=\\frac{1}{\\simplify{2*{k}}(x^2+y^2)}dudv\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\implies \\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k2}(x^2+y^2)\\frac{1}{\\simplify{2*{k}}(x^2+y^2)}dudv=\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\frac{\\var{k2}}{\\simplify{2*{k}}}dudv\\)</p>\n<p></p>\n<p>Inner integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a1}^\\var{b1}\\frac{\\var{k2}}{\\simplify{2*{k}}}du\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{k2}}{\\simplify{2*{k}}}u\\big|_\\var{a1}^\\var{b1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{k2}}{\\simplify{2*{k}}}(\\var{b1})-\\frac{\\var{k2}}{\\simplify{2*{k}}}(\\var{a1})\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\var{inner}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Outer integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a2}^\\var{b2}\\var{inner}dv\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\var{inner}v\\big|_\\var{a2}^\\var{b2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\var{inner}(\\var{b2})-\\var{inner}(\\var{a2})\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{inner}*({b2}-{a2})}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "functions": {}, "preamble": {"js": "", "css": ""}, "extensions": [], "ungrouped_variables": ["a2", "a1", "b1", "b2", "k", "k2", "inner"], "variables": {"k": {"name": "k", "definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "inner": {"name": "inner", "definition": "{k2}*({b1}-{a1})/(2*{k})", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a1": {"name": "a1", "definition": "random(0..3#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b2": {"name": "b2", "definition": "random(5..7#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b1": {"name": "b1", "definition": "random(4..8#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k2": {"name": "k2", "definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "a2": {"name": "a2", "definition": "random(0..4#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}}, "rulesets": {}, "name": "Change of variables", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "tags": [], "parts": [{"marks": 0, "type": "gapfill", "gaps": [{"maxValue": "{k2}*({b1}-{a1})*({b2}-{a2})/(2*{k})", "type": "numberentry", "precision": "3", "mustBeReduced": false, "showCorrectAnswer": true, "precisionPartialCredit": "75", "showFeedbackIcon": true, "correctAnswerFraction": false, "scripts": {}, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "marks": 1, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "minValue": "{k2}*({b1}-{a1})*({b2}-{a2})/(2*{k})", "variableReplacements": []}], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "<p>Insert your answer correct to three decimal places</p>\n<p>Answer =&nbsp;[[0]]</p>", "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}